2018/10/11 by Yuuki Shimizu, Shimizu, Yuuki
Engineering · Mathematics · #35Q31 #53C21 #58J05 #76B47 #Differential Equations and Boundary Problems #Differential Geometry (math.DG) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.1810.09523
openalex publication_date 2018/10/11 · openalex created_date 2022/08/02 · openalex updated_date 2026/07/28
We derive an analytic formula for the hydrodynamic Green function and the Robin function on every orientable surface admitting a hydrodynamic Killing vector field. Closed-form expressions are provided for all fourteen canonical Riemann surfaces, covering both compact and non-compact cases; the formulae satisfy the slip boundary condition and generate complete Hamiltonian vector fields. As an application, we clarify the mechanism whereby the curvature affects a point vortex in both qualitative and quantitative viewpoints. Qualitatively, we show a single point vortex is governed by a Hamiltonian flow whose vorticity is given by the curvature up to area constant. Quantitatively, on a rectangular torus with periodic curvature we use the analytic formula to describe two regimes: linear response that mirrors the curvature wave when the mean component is small, and a nonlinear response with amplitude resonance. The results supply a unified tool for detailed studies of point vortex dynamics and Euler-Arnold flows on surfaces with continuous symmetry.